Solve for $x$ : $ 8|x + 7| + 4 = -6|x + 7| + 6 $
Answer: Add $ {6|x + 7|} $ to both sides: $ \begin{eqnarray} 8|x + 7| + 4 &=& -6|x + 7| + 6 \\ \\ { + 6|x + 7|} && { + 6|x + 7|} \\ \\ 14|x + 7| + 4 &=& 6 \end{eqnarray} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} 14|x + 7| + 4 &=& 6 \\ \\ { - 4} &=& { - 4} \\ \\ 14|x + 7| &=& 2 \end{eqnarray} $ Divide both sides by ${14}$ $ \dfrac{14|x + 7|} {{14}} = \dfrac{2} {{14}} $ Simplify: $ |x + 7| = \dfrac{1}{7}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 7 = -\dfrac{1}{7} $ or $ x + 7 = \dfrac{1}{7} $ Solve for the solution where $x + 7$ is negative: $ x + 7 = -\dfrac{1}{7} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& -\dfrac{1}{7} \\ \\ {- 7} && {- 7} \\ \\ x &=& -\dfrac{1}{7} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $7$ $ x = - \dfrac{1}{7} {- \dfrac{49}{7}} $ $ x = -\dfrac{50}{7} $ Then calculate the solution where $x + 7$ is positive: $ x + 7 = \dfrac{1}{7} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& \dfrac{1}{7} \\ \\ {- 7} && {- 7} \\ \\ x &=& \dfrac{1}{7} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $7$ $ x = \dfrac{1}{7} {- \dfrac{49}{7}} $ $ x = -\dfrac{48}{7} $ Thus, the correct answer is $x = -\dfrac{50}{7} $ or $x = -\dfrac{48}{7} $.